Owing to the estimates of Lemma REF  and using an approximation argument similar to [1]} Remark 2.3 we see that any minimizer of the relaxed problem REF  satisfies the Fokker-Planck equation against any function \(\phi \in \mathbb {A}^{\prime }\) . Now we define \(\mathbb {B}\)  to be the set of tuples \((m,\omega ,W,n)\)  in \(\mathcal {M}^+([0,T]\times \mathbb {R}^d) \times \mathcal {M}([0,T]\times \mathbb {R}^d,\mathbb {R}^d) \times \mathcal {M}([0,T]\times \mathbb {R}^d,\mathbb {S}_d(\mathbb {R})) \times \mathcal {M}^+(\mathbb {R}^d) \)  such that \(\omega \) , \(W\)  are absolutely continuous with respect to \(m\) . \(J_{RP}^{\prime }\)  is defined on \(\mathbb {B}\)  by
\(J_{RP}^{\prime }(m,\omega ,W,n) = \int _0^T \int _{\mathbb {R}^d} \left[ H_1^* \left( x,\frac{d\omega }{dm}(t,x) \right) + H_2^* \left(x, \frac{dW}{dm}(t,x) \right) \right] m(dt,dx) + \int _{\mathbb {R}^d} g(x) n(dx). \) 
